Problem 86 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{3}{2}-\frac{2}{3}$$

Step-by-Step Solution

Answer

The result of \(\frac{3}{2} - \frac{2}{3}\) is \(\frac{5}{6}\).

Step 1: Finding the Least Common Multiple

Find the least common multiple (LCM) of the denominators 2 and 3. The LCM of 2 and 3 is 6.

Step 2: Adjusting the fractions

Change both fractions to have the common denominator. Mutiply the first fraction's numerator and denominator by 3, and for the second fraction multiply by 2. This yields the fractions as: \(\frac{3*3}{2*3} - \frac{2*2}{3*2}\) or \(\frac{9}{6} - \frac{4}{6}\).

Step 3: Perform Subtraction Operation

Subtract the numerators (\(9 - 4\)) while keeping the same denominator (6). This results in: \(\frac{9 - 4}{6} = \frac{5}{6}\).

Step 4: Simplify the Result

If possible, simplify the resulting fraction. In this case, \(\frac{5}{6}\) is already at its simplest form as 5 and 6 don't have common divisors except 1.

Key Concepts

Least Common Multiple (LCM)

Understanding how to find the Least Common Multiple (LCM) is crucial when working with fractions, especially in subtracting and adding them. The LCM of two numbers is the smallest number that is a multiple of both.

Here's an easy way to find the LCM: List the multiples of each number until you find the first common multiple. For instance, with the numbers 2 and 3, the multiples of 2 are 2, 4, 6, 8, and so on, while the multiples of 3 are 3, 6, 9, and so forth. The first common multiple is 6, hence, the LCM of 2 and 3 is 6. We need the LCM to get to a common denominator which is essential for fraction operations.

Common Denominator

A common denominator refers to a shared denominator between two or more fractions. When subtracting or adding fractions, they need to have the same denominator. This is important because it allows us to compare or combine fractions without changing their values.

To adjust fractions to have a common denominator, simply multiply the numerator and the denominator of each fraction by a number that will result in the LCM. For the fractions \(\frac{3}{2}\) and \(\frac{2}{3}\), we multiply the first by 3 and the second by 2, giving us \(\frac{9}{6}\) and \(\frac{4}{6}\) respectively, both now with a common denominator of 6.

Simplifying Fractions

The process of simplifying fractions is to reduce them to their most basic form. This means that the numerator and denominator have no common divisors other than 1.

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. If the GCD is 1, the fraction is already in its simplest form. As in our exercise, the fraction \(\frac{5}{6}\) cannot be simplified further since 5 and 6 are relatively prime—that is, they have no common factors other than 1.